Theorem al0ssb 5314

Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
al0ssb	⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Distinct variable group:   𝑦,𝑋

Proof of Theorem al0ssb

Step	Hyp	Ref	Expression
1		0ex 5313	. . 3 ⊢ ∅ ∈ V
2		sseq2 4022	. . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅))
3		ss0b 4407	. . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅)
4	2, 3	bitrdi 287	. . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅))
5	1, 4	spcv 3605	. 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅)
6		0ss 4406	. . . 4 ⊢ ∅ ⊆ 𝑦
7	6	ax-gen 1792	. . 3 ⊢ ∀𝑦∅ ⊆ 𝑦
8		sseq1 4021	. . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
9	8	albidv 1918	. . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦))
10	7, 9	mpbiri 258	. 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦)
11	5, 10	impbii 209	1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Syntax hints:   ↔ wb 206  ∀wal 1535   = wceq 1537   ⊆ wss 3963  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340

This theorem is referenced by:  iota0def  46988  aiota0def  47046